Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{7p^2 + 63p - 70}{-2p^3 - 2p^2 + 180p}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {7(p^2 + 9p - 10)} {-2p(p^2 + p - 90)} $ $ z = -\dfrac{7}{2p} \cdot \dfrac{p^2 + 9p - 10}{p^2 + p - 90} $ Next factor the numerator and denominator. $ z = - \dfrac{7}{2p} \cdot \dfrac{(p + 10)(p - 1)}{(p + 10)(p - 9)}$ Assuming $p \neq -10$ , we can cancel the $p + 10$ $ z = - \dfrac{7}{2p} \cdot \dfrac{p - 1}{p - 9}$ Therefore: $ z = \dfrac{ -7(p - 1)}{ 2p(p - 9)}$, $p \neq -10$